When an object falls freely near Earth's surface, its kinetic energy increases as it accelerates due to gravity. The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the object and \( v \) is its velocity.
During free fall, the velocity of the object increases linearly with time due to the constant acceleration of gravity (approximately \( 9.81 , \text{m/s}^2 \)):
\[ v = gt \]
Substituting this relationship into the kinetic energy equation, we get:
\[ KE = \frac{1}{2} m (gt)^2 = \frac{1}{2} mg^2 t^2 \]
This indicates that the kinetic energy of the falling object is proportional to the square of the time elapsed (\( t^2 \)). As time progresses, the kinetic energy increases quadratically.
In a graph representing kinetic energy versus time, you would see a curve that starts from the origin and curves upward, illustrating the quadratic relationship. The graph would not be linear; instead, it would show that as more time elapses, the increase in kinetic energy becomes steeper.
Thus, the best representation of the relationship between the object’s kinetic energy and time elapsed would be a parabolic graph opening upwards, indicating a quadratic relationship.